Results 1 to 5 of 5

Math Help - Join Complete Sufficient Statistic

  1. #1
    Member
    Joined
    Oct 2009
    Posts
    218

    Join Complete Sufficient Statistic

    Let X1,...,Xn be from a N(\mu,4 \sigma^2)

    A) Find the Joint complete sufficient statistics for \mu and 4 \sigma^2

    B)Find an MVUE for 4 \sigma^2

    C)Find an MVUE for \mu^2

    a)From a previous problem we did in class, I know Y_1 = \Sigma X^2_i and Y_2 = \Sigma X_i are Joint complete sufficient statistics for \mu and 4 \sigma^2

    b)Let Z_2 = \frac{Y_1 - Y^2_2/n}{n-1} = \frac{\Sigma(X_i - \overline{X})^2}{n-1} which is the sample variance.

    Would E(Z_2) = 4 \sigma^2 or would it be E(Z_2) = \sigma^2

    c)Let Z_1 = \frac{Y_2}{n} = \overline{X} and E(Z_1)=\mu

    E(Z^2_1) = E(Z_1)^2 + Var(Z_1) = \mu^2 +\frac{4 \sigma^2}{n}

    Therefore Z^2_1 - \frac{Z_2}{n} is an MVUE for \mu^2 according to Lehmann and Scheffe Theroem.

    Did I make a mistake somewhere, and would E(Z_2) = 4 \sigma^2 or would it be E(Z_2) = \sigma^2
    Last edited by statmajor; February 24th 2010 at 04:52 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    You left the ^ off your Z2

    Quote Originally Posted by statmajor View Post
    Let X1,...,Xn be from a N(\mu,4 \sigma^2)

    A) Find the Joint complete sufficient statistics for \mu and 4 \sigma^2

    B)Find an MVUE for 4 \sigma^2

    C)Find an MVUE for \mu^2

    a)From a previous problem we did in class, I know Y_1 = \Sigma X^2_i and Y_2 = \Sigma X_i are Joint complete sufficient statistics for \mu and 4 \sigma^2

    b)Let Z_2 = \frac{Y_1 - Y^2_2/n}{n-1} = \frac{\Sigma(X_i - \overline{X})2}{n-1} which is the sample variance.

    Would E(Z_2) = 4 \sigma^2 or would it be E(Z_2) = \sigma^2

    c)Let Z_1 = \frac{Y_2}{n} = \overline{X} and E(Z_1)=\mu

    E(Z^2_1) = E(Z_1)^2 + Var(Z_1) = \mu^2 +\frac{4 \sigma^2}{n}

    Therefore Z^2_1 - \frac{Z_2}{n} is an MVUE for \mu^2 according to Lehmann and Scheffe Theroem.

    Did I make a mistake somewhere, and would E(Z_2) = 4 \sigma^2 or would it be E(Z_2) = \sigma^2
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2009
    Posts
    218
    So would would E(Z_2) = 4 \sigma^2?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    yes, E(Z_2) = V(X_1)=4 \sigma^2
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Oct 2009
    Posts
    218
    Thank you.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. sufficient statistic
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: January 15th 2012, 03:12 AM
  2. Sufficient statistic
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: February 14th 2011, 10:21 AM
  3. Finding the Complete Sufficient Statistic
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: January 21st 2010, 06:09 PM
  4. Finding a sufficient statistic
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: October 5th 2009, 09:52 PM
  5. Join of complete r-partite graphs
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: January 12th 2009, 12:08 AM

Search Tags


/mathhelpforum @mathhelpforum